作者tsungjen (湛红)
看板trans_math
标题Re: 请益
时间Fri Jun 2 18:05:41 2006
※ 引述《diedheart (die)》之铭言：
: find extreme value of
: f(x)=e -xy (-xy是在次方...即右上角)
: x^2+4y^2<=1
: 谢谢
f(x,y)=e^(-xy)
g(x,y)=x^2+4y^2
fx=-ye^(-xy)
fy=-xe^(-xy)
gx=2x
gy=8y
By Lagrange Multiplier Thm
fx=a gx => -ye^(-xy)=2ax
fy=a gy => -xe^(-xy)=8ay
(1)x^2+4y^2=1
y x x= 2^(-1/2),-2^(-1/2)
a= - ----e^(-xy) = - ----e^(-xy) => x^2=4y^2 => y= 2^(-1/2),-2^(-1/2)
2x 8y
fmax=e^(1/2)
fmin=e^(-1/2)
(2)x^2+4y^2<1
solve critical points fx=0 and fy=0
fx=-ye^(-xy)=0 => y=0 => fxx=y^2e^(-xy)=0 and fxy=fyx=(xy-1)e^(-xy)=-1
fy=-xe^(-xy)=0 => x=0 => fyy=x^2e^(-xy)=0
/ fxx fxy /
d= / / = -1 < 0 => saddle point
/ fyx fyy /
(行列式)
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.114.217.21
1F：推 diedheart:谢谢你 06/03 21:42